Eight people were taken to hospital after the attack outside the mosque, which is also a community centre.
A 48-year-old man has been arrested on suspicion of attempted murder.
A group of people were helping a man who had collapsed when they were hit by the van just after midnight. The man has died, but it is unclear whether this was as a result of the attack.
This is the fourth terror attack in the UK in four months, after incidents in Westminster, Manchester and on London Bridge.
Police said all the victims of the attack, which was in the area of Finsbury Park Mosque, were Muslim.
Mrs May was speaking after chairing a meeting of the government's emergency Cobra committee.
She said the driver of the van acted alone, adding: "Hatred and evil of this kind will never succeed."
She also said that police declared it a terrorist incident within eight minutes, and that it was "an attack on Muslims near their place of worship".
Mrs May added that "there has been far too much tolerance of extremism over many years".
"It is a reminder that terrorism, extremism and hatred take many forms; and our determination to tackle them must be the same whoever is responsible."
Eyewitness Abdul Rahman said the driver had said he wanted to "kill all Muslims" and told the BBC he struck the man and helped subdue him.
Mr Rahman said: "When the guy came out from his van he wanted to escape, run away and he was saying 'I want to kill Muslims. 'I want to kill Muslims.'
"I hit him on his stomach... and then me and the other guys... we held him to the ground until he couldn't move. We stopped him until the police came."
Metropolitan Police's deputy assistant commissioner Neil Basu said the terror attack began when the van was driven into a man who was already being given first aid by the public on the pavement.
Several of those in hospital are "seriously injured" and there are currently no other suspects, he added.
Earlier, police also said the arrested man would be the "subject of a mental health assessment in due course".
Extra police are being deployed to reassure communities, especially those observing Ramadan, Mayor of London Sadiq Khan said.
He also said his "thoughts and prayers" were with those affected by the "horrific terrorist attack on innocent people".
"While this appears to be an attack on a particular community, like the terrible attacks in Manchester, Westminster and London Bridge it is also an assault on all our shared values of tolerance, freedom and respect."
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